Specify the antiderivative of the function f (x) = 6x, the graph of which passes through the point M (-1; 5).

To find the antiderivative, one should do the reverse operations of taking the derivative. If the derivative of a variable to the power of n is equal to the product of the number n by the variable to the power reduced by one (n – 1), then in the case of finding the antiderivative, the variable should be raised to a power by one greater, and the coefficient must be divided by a number equal to the increased power of the variable. Then, taking into account the fact that the derivative of the number (const) is always zero, and the derivative is:

f (x) = 6 * x;

The antiderivative is equal to:

F (x) = 6 * x ^ 2/2 + const = 3 * x2 + const.

It remains to find const, which will ensure the passage of the graph through the point M (-1; 5). To do this, we equate the function 5, x will be considered equal to -1, and we will denote const by the letter C and solve the resulting equation with respect to the variable C:

3 * x2 + const = 3 * (-1) 2 + C = 5;

3 + C = 5;

C = 5 – 3 = 2;

Thus, the free term is found and the antiderivative is:

F (x) = 3 * x ^ 2 + 2.